While I am nowhere near being a master story teller, I found joining Toastmasters to be the best thing I have done in my life (wondering why I left it so late, leaves me slapping my self each time). Toastmasters (like many other programs that teach communication/presentation skills) is ultimately a study of technique and story telling needs truck loads of it to be engaging and effective. If TM can help a stuttering introvert with chronic low self-esteem like myself, imagine what it can do for you!
I would add that it is really worth seeing professional storytellers, especially if there is a storytelling festival in your area. These are often people who tell both traditional folk tales and personal stories, and sometimes just crazy invented stuff. It's not comedy but it's often funny, and it's not theatre but it's often dramatic. You can be in a tent with 2000 people and feel like you just had a great conversation with an old friend. There is a ton of wisdom, practice, and stagecraft involved in getting to that level, but at the same time the fundamentals are simple and consistent: Respect the audience, respect the story, and let some of your own personality in.
The best advice in the world only works in the context of practice.
Practicing makes advice meaningful.
Practice and then practice more. Then practice more.
Get feedback. Seek advice. Iterate. But practice.
Name anyone who mastered a performance art who didn't practice more than most. Even if you can, I can name a hundred who did.
Practice, practice, practice.
Find audiences you can practice with and practice with them. Until then, practice by yourself, in front of a wall if that's what you have. Use a mirror or a camera. Practice. The more you practice, the more audiences you'll find.
One illusion we have is that a great storyteller is great the first time with no practice or anything.
Hangout with someone who tells good stories and you will hear the same story come back often. The more they tell it, the more it sounds like it is the very first time they tell it.
I watched a lot of videos from Seth Godin, even the jokes are the exact same, but everytime the audience is hearing it for the first time.
How to deal with people who tend to re-tell the same stories all the time? Someone very close to me keeps telling me the same stories every time I meet them (a few times per year) and it's very annoying. Sometimes I kind of finish their story as soon as they start telling it and I feel like this has been the least offensive, but I can't do that for like 3 stories in a row.
I can't tell them to just stop. They really like telling these stories. But if it's a 5-minute story and I've already heard it 5 times before, it's just not fun at all.
I agree with other comments that practice is important.
Even more important is getting good input that you can model. Read fiction every day. While non-fiction is generally read for the purpose of furthering an instrumental goal (e.g. learning how to do something or learning about something), fiction's one job is to be interesting. As a result, even storytellers who are relatively weak by the standards of fiction authors are well ahead of typical business speakers or writers.
Pay attention to what keeps your attention and why. Pay attention to what you remember a week after you've finished the book.
A lot of practical advice in the comments so far. Practice is important. Don't wait in your story to hint at the outcome. Don't give the punch line until it is time but entice the audience with what is coming early on. Hook them.
Like all skills, story telling is something that improves with practice. As you do it more often, you tend to think about it more often and improve your skills. I used to discuss everyday news with my sister. She is a super busy person and liked listening to news through me.So I finally decided to build a startup out of it. I am trying to present everyday news in the form of a story. It has been almost a month since I have started writing news every day in the form of a story and I believe it keeps getting better with time.
* http://witty.news/ is my site. The news is mostly for Indian audience.
I recommend reading George Campbell's "The Hero with a Thousand Faces".
This is about comparative mythology, aka comparative story telling.
It opened my mind to be able to understand all kinds of storys on another level, so I guess it is also helpful for writing those.
Two aspects: performance and content. For performance I concur with the other folks advice here to practice a lot. For content I recommend reading Story - by Robert McKee. There's lots of knowledge on how to hold an audience's attention.
Practice is important and when coupled with introspection/self-reflection can help a lot in identifying areas where easy improvement can be developed. However, of equal importance is targeted study. If you want to tell better jokes, study successful comedians and their jokes. If you want to write, read stories by successful writers. Meanwhile, as you do these things always try to understand why the audiences for these things liked them so much.
I think the best advice I can give is the nike approach: Just Do It. If you want to be good at telling stories then you have to drop your ego and just tell them. I'm a blogger and blogging is the best thing that ever happened to my story telling. Being able to write whenever you want, without a gatekeeper, editor or schedule meant that my story telling skills could develop at their own pace.
I don't buy that 80/20 breakdown. Timing and structure can be different in two tellings of the same story, and those have little to do with the in-person attributes of a storyteller. A blogger still has to draw attention, play with the audience's emotions and anticipation, etc. So there's an obvious difference between story telling and blogging, but it may not be as substantial as neglecting 80% of the whole deal.